Time as a Quantum Observable, Canonically Conjugated to Energy, and Foundations of Self-Consistent Time Analysis of Quantum Processes

<table>The dependences of <svg height="15.3625" id="M574" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 83.112503 15.3625" width="83.112503" xmlns="http://www.w3.org/2000/svg">
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<g transform="translate(72,-59.71)">
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<tspan style="font-size: 12.50px; " x="0" y="0">Δ</tspan>
<tspan style="font-size: 12.50px; " x="9.0271664" y="0">𝑘</tspan>
<tspan style="font-size: 12.50px; " x="19.267122" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="31.307512" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="37.559013" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="40.684761" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="46.93626" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="57.351261" y="0">Å</tspan>
</text>
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</svg>, <svg height="15.3625" id="M575" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 83.112503 15.3625" width="83.112503" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,15.3625)">
<g transform="translate(72,-59.71)">
<text transform="matrix(1,0,0,-1,-71.95,59.93)">
<tspan style="font-size: 12.50px; " x="0" y="0">Δ</tspan>
<tspan style="font-size: 12.50px; " x="9.0271664" y="0">𝑘</tspan>
<tspan style="font-size: 12.50px; " x="19.267122" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="31.307512" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="37.559013" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="40.684761" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="46.93626" y="0">5</tspan>
<tspan style="font-size: 12.50px; " x="57.351261" y="0">Å</tspan>
</text>
</g>
</g>
</svg> and <svg height="15.3625" id="M576" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 75.300003 15.3625" width="75.300003" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,15.3625)">
<g transform="translate(72,-59.71)">
<text transform="matrix(1,0,0,-1,-71.95,59.93)">
<tspan style="font-size: 12.50px; " x="0" y="0">Δ</tspan>
<tspan style="font-size: 12.50px; " x="9.0271664" y="0">𝑘</tspan>
<tspan style="font-size: 12.50px; " x="19.267122" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="31.307512" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="37.559013" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="40.684761" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="51.099762" y="0">Å</tspan>
</text>
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</svg> from <i>a</i>.</table>

Advances in Mathematical Physics

fig4

Figure 4

Figure 4: Time as a Quantum Observable, Canonically Conjugated to Energy, and Foundations of Self-Consistent Time Analysis of Quantum Processes 